Subgroups — C₅×#26190 #11948910

Order 1280.

766 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3045	C₂	1	1	yes	maximal, characteristic
3	C₅×#2951	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2913	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2093	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2078	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2037	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1931	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1922	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1907	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1883	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1876	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1859	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1859	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
17	C₂×C₅×#135	C₂²	1	1	yes	characteristic
18	C₂×C₅×#131	C₂²	1	1	yes	characteristic
19	C₅×#535	C₂²	1	1	yes	characteristic
20	C₅×#520	C₂²	1	1	yes	characteristic
21	C₅×#520	C₂²	1	1	yes	characteristic
22	C₅×#520	C₂²	1	1	yes	characteristic
23	C₅×#482	—	2	1	no	—
24	C₅×#482	—	2	1	no	—
25	C₅×#481	—	2	1	no	—
26	C₅×#480	—	2	1	no	—
27	C₅×#464	—	2	1	no	—
28	C₅×#463	—	2	1	no	—
29	C₅×#460	—	2	1	no	—
30	C₅×#459	—	2	1	no	—
31	C₅×#440	—	2	1	no	—
32	C₅×#440	—	2	1	no	—
33	C₄×C₅×Q₁₆	—	2	1	no	—
34	C₄×C₅×D₈	—	2	1	no	—
35	C₅×#428	—	2	1	no	—
36	C₅×#428	—	2	1	no	—
37	C₅×#428	C₂²	1	1	yes	characteristic
38	C₅×#428	C₂²	1	1	yes	characteristic
39	C₅×#428	C₂²	1	1	yes	characteristic
40	C₅×#427	—	2	1	no	—
41	C₅×#427	C₂²	1	1	yes	characteristic
42	C₂×C₅×#107	—	2	1	no	—
43	C₂×C₅×#107	C₂²	1	1	yes	characteristic
44	C₂×C₅×#107	C₂²	1	1	yes	characteristic
45	C₅×#424	C₂²	1	1	yes	characteristic
46	C₂×C₅×#104	C₂²	1	1	yes	characteristic
47	C₂×C₅×#104	C₂²	1	1	yes	characteristic
48	C₅×#418	C₂²	1	1	yes	characteristic
49	C₅×#417	C₂²	1	1	yes	characteristic
50	C₅×#417	C₂²	1	1	yes	characteristic
51	C₅×#417	C₂²	1	1	yes	characteristic
52	C₂×C₅×#102	C₂²	1	1	yes	characteristic
53	C₂×C₅×#102	C₂²	1	1	yes	characteristic
54	C₂×C₅×#100	C₂²	1	1	yes	characteristic
55	C₅×#408	C₂²	1	1	yes	characteristic
56	C₅×#405	C₂²	1	1	yes	characteristic
57	C₅×#390	C₂²	1	1	yes	characteristic
58	C₅×#384	C₂²	1	1	yes	characteristic
59	C₅×#343	C₂²	1	1	yes	characteristic
60	C₅×#343	C₂²	1	1	yes	characteristic
61	C₅×#343	C₂²	1	1	yes	characteristic
62	C₅×#341	C₂²	1	1	yes	characteristic
63	C₅×#341	C₂²	1	1	yes	characteristic
64	C₅×#339	C₂²	1	1	yes	characteristic
65	C₅×#339	C₂²	1	1	yes	characteristic
66	C₅×#336	C₂²	1	1	yes	characteristic
67	#26190	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	C₂³	1	1	yes	characteristic
69	C₂×C₅×#41	C₂³	1	1	yes	characteristic
70	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
71	C₅×AGL(1,ℤ/8)	—	2	2	no	—
72	C₅×AGL(1,ℤ/8)	—	2	2	no	—
73	C₅×#135	—	2	2	no	—
74	C₅×#135	—	2	2	no	—
75	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
76	C₂×C₅×SD₁₆	—	2	1	no	—
77	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
78	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
79	C₂×C₅×D₈	—	2	1	no	—
80	C₂×C₅×D₈	D₄	1	1	yes	characteristic
81	C₅×#131	—	2	2	no	—
82	C₅×#131	—	2	2	no	—
83	C₂×C₅×M₁₆	—	2	1	no	—
84	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
85	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
86	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
88	C₂²×C₅×C₈	—	2	1	no	—
89	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
90	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
91	C₅×#127	—	2	1	no	—
92	C₅×#124	—	2	1	no	—
93	C₅×#121	—	2	1	no	—
94	C₅×#121	—	2	1	no	—
95	C₅×#120	—	4	1	no	—
96	C₄×C₅×Q₈	—	2	1	no	—
97	C₄×C₅×Q₈	—	2	1	no	—
98	C₄×C₅×Q₈	—	2	1	no	—
99	C₄×C₅×D₄	—	2	1	no	—
100	C₄×C₅×D₄	—	2	1	no	—
101	C₄×C₅×D₄	—	2	1	no	—
102	C₄×C₅×D₄	—	4	1	no	—
103	C₄×C₅×D₄	—	4	1	no	—
104	C₄×C₅×D₄	—	4	1	no	—
105	C₅×#117	—	2	1	no	—
106	C₅×#117	—	2	1	no	—
107	C₅×#117	—	2	1	no	—
108	C₅×#117	C₂³	1	1	yes	characteristic
109	C₅×#117	C₂³	1	1	yes	characteristic
110	C₂×C₅×C₄⋊C₄	—	2	1	no	—
111	C₂×C₅×C₄⋊C₄	—	2	1	no	—
112	C₂×C₅×C₄⋊C₄	—	2	1	no	—
113	C₂×C₅×C₄⋊C₄	—	4	1	no	—
114	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
115	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
116	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
117	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
118	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
119	C₅×#107	—	2	1	no	—
120	C₅×#107	—	2	1	no	—
121	C₅×#107	—	2	1	no	—
122	C₅×#107	—	2	1	no	—
123	C₅×#107	—	2	1	no	—
124	C₅×#107	—	2	1	no	—
125	C₅×#107	—	2	1	no	—
126	C₅×#107	D₄	1	1	yes	characteristic
127	C₅×#107	D₄	1	1	yes	characteristic
128	C₅×#106	—	2	1	no	—
129	C₅×#106	—	2	1	no	—
130	C₅×#106	—	2	1	no	—
131	C₅×#106	D₄	1	1	yes	characteristic
132	C₅×#106	D₄	1	1	yes	characteristic
133	C₅×C₄⋊C₈	—	2	1	no	—
134	C₅×C₄⋊C₈	—	2	1	no	—
135	C₅×#104	—	2	2	no	—
136	C₅×#104	—	2	2	no	—
137	C₅×#103	—	2	1	no	—
138	C₅×#103	—	2	1	no	—
139	C₅×#103	—	2	1	no	—
140	C₅×#102	—	2	1	no	—
141	C₅×#102	—	2	1	no	—
142	C₅×#102	—	2	1	no	—
143	C₅×#102	—	2	1	no	—
144	C₅×#102	—	2	1	no	—
145	C₅×#102	—	2	1	no	—
146	C₅×#102	—	2	1	no	—
147	C₅×#102	—	2	1	no	—
148	C₅×#102	—	2	1	no	—
149	C₅×#100	—	2	2	no	—
150	C₅×C₂²⋊C₈	—	2	1	no	—
151	C₅×C₂²⋊C₈	—	2	1	no	—
152	C₅×#97	—	2	1	no	—
153	C₄×C₅×C₈	—	2	1	no	—
154	#3096	C₂×C₅	1	1	yes	characteristic, complemented
155	#3045	C₂×C₅	1	1	yes	characteristic
156	#2951	C₂×C₅	1	1	yes	characteristic, complemented
157	#2913	C₂×C₅	1	1	yes	characteristic, complemented
158	#2093	C₂×C₅	1	1	yes	characteristic, complemented
159	#2089	C₂×C₅	1	1	yes	characteristic, complemented
160	#2078	C₂×C₅	1	1	yes	characteristic, complemented
161	#2037	C₂×C₅	1	1	yes	characteristic, complemented
162	#1931	C₂×C₅	1	1	yes	characteristic, complemented
163	#1922	C₂×C₅	1	1	yes	characteristic, complemented
164	#1907	C₂×C₅	1	1	yes	characteristic, complemented
165	#1883	C₂×C₅	1	1	yes	characteristic, complemented
166	#1876	C₂×C₅	1	1	yes	characteristic, complemented
167	#1859	C₂×C₅	1	1	yes	characteristic, complemented
168	#1859	C₂×C₅	1	1	yes	characteristic, complemented
169	C₂⁴×C₅	—	2	1	no	—
170	C₅×#41	—	2	1	no	—
171	C₅×#41	—	2	1	no	—
172	C₅×#41	—	2	1	no	—
173	C₅×#41	—	2	1	no	—
174	C₅×#41	—	2	1	no	—
175	C₅×#41	—	2	1	no	—
176	C₅×#41	#41	1	2	yes	—
177	C₅×#41	#41	1	2	yes	—
178	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
179	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
180	C₂×C₅×D₄	—	2	1	no	—
181	C₂×C₅×D₄	—	2	1	no	—
182	C₂×C₅×D₄	—	2	1	no	—
183	C₂×C₅×D₄	—	2	1	no	—
184	C₂×C₅×D₄	—	2	1	no	—
185	C₂×C₅×D₄	—	2	1	no	—
186	C₂×C₅×D₄	—	2	1	no	—
187	C₂×C₅×D₄	—	2	1	no	—
188	C₂×C₅×D₄	—	4	1	no	—
189	C₂×C₅×D₄	—	4	1	no	—
190	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
191	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₅	—	2	1	no	—
193	C₂²×C₄×C₅	—	2	1	no	—
194	C₂²×C₄×C₅	—	2	1	no	—
195	C₂²×C₄×C₅	—	2	1	no	—
196	C₂²×C₄×C₅	—	2	1	no	—
197	C₂²×C₄×C₅	—	2	1	no	—
198	C₂²×C₄×C₅	—	4	1	no	—
199	C₂²×C₄×C₅	—	4	1	no	—
200	C₂²×C₄×C₅	—	4	1	no	—
201	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
202	C₅×Q₁₆	—	2	2	no	—
203	C₅×SD₁₆	—	2	2	no	—
204	C₅×SD₁₆	—	2	2	no	—
205	C₅×SD₁₆	—	4	2	no	—
206	C₅×D₈	—	2	2	no	—
207	C₅×D₈	—	4	2	no	—
208	C₅×M₁₆	—	2	4	no	—
209	C₅×M₁₆	—	2	1	no	—
210	C₅×M₁₆	—	2	1	no	—
211	C₅×M₁₆	—	2	1	no	—
212	C₅×M₁₆	—	2	1	no	—
213	C₅×M₁₆	—	2	1	no	—
214	C₅×M₁₆	—	2	1	no	—
215	C₅×M₁₆	—	2	1	no	—
216	C₅×M₁₆	#41	1	2	yes	—
217	C₂×C₅×C₈	—	2	4	no	—
218	C₂×C₅×C₈	—	2	1	no	—
219	C₂×C₅×C₈	—	2	1	no	—
220	C₂×C₅×C₈	—	2	1	no	—
221	C₂×C₅×C₈	—	2	1	no	—
222	C₂×C₅×C₈	—	2	1	no	—
223	C₂×C₅×C₈	—	2	1	no	—
224	C₂×C₅×C₈	—	2	1	no	—
225	C₂×C₅×C₈	—	2	1	no	—
226	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
227	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
228	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
229	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
230	C₅×C₄⋊C₄	—	2	1	no	—
231	C₅×C₄⋊C₄	—	2	1	no	—
232	C₅×C₄⋊C₄	—	2	1	no	—
233	C₅×C₄⋊C₄	—	2	1	no	—
234	C₅×C₄⋊C₄	—	2	1	no	—
235	C₅×C₄⋊C₄	—	2	1	no	—
236	C₅×C₄⋊C₄	—	4	1	no	—
237	C₅×C₄⋊C₄	—	4	1	no	—
238	C₅×C₄⋊C₄	—	4	1	no	—
239	C₅×C₄⋊C₄	—	8	2	no	—
240	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
241	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
242	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
243	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
244	C₅×C₂²⋊C₄	—	2	1	no	—
245	C₅×C₂²⋊C₄	—	2	1	no	—
246	C₅×C₂²⋊C₄	—	4	1	no	—
247	C₅×C₂²⋊C₄	—	4	1	no	—
248	C₅×C₂²⋊C₄	—	4	1	no	—
249	C₅×C₂²⋊C₄	—	4	1	no	—
250	C₅×C₂²⋊C₄	—	4	1	no	—
251	C₅×C₂²⋊C₄	—	8	2	no	—
252	C₄²×C₅	—	2	1	no	—
253	C₄²×C₅	—	2	1	no	—
254	C₄²×C₅	—	2	1	no	—
255	C₄²×C₅	—	2	1	no	—
256	C₄²×C₅	—	4	1	no	—
257	C₄²×C₅	—	4	1	no	—
258	C₄²×C₅	—	4	1	no	—
259	C₂×AGL(1,ℤ/8)	C₂²×C₅	1	1	yes	characteristic
260	C₂×#135	C₂²×C₅	1	1	yes	characteristic
261	C₂×#131	C₂²×C₅	1	1	yes	characteristic
262	#535	C₂²×C₅	1	1	yes	characteristic
263	#520	C₂²×C₅	1	1	yes	characteristic
264	#520	C₂²×C₅	1	1	yes	characteristic
265	#520	C₂²×C₅	1	1	yes	characteristic
266	#482	—	2	1	no	—
267	#482	—	2	1	no	—
268	#481	—	2	1	no	—
269	#480	—	2	1	no	—
270	#464	—	2	1	no	—
271	#463	—	2	1	no	—
272	#460	—	2	1	no	—
273	#459	—	2	1	no	—
274	#440	—	2	1	no	—
275	#440	—	2	1	no	—
276	C₄×Q₁₆	—	2	1	no	—
277	C₄×D₈	—	2	1	no	—
278	#428	—	2	1	no	—
279	#428	—	2	1	no	—
280	#428	C₂²×C₅	1	1	yes	characteristic
281	#428	C₂²×C₅	1	1	yes	characteristic
282	#428	C₂²×C₅	1	1	yes	characteristic
283	#427	—	2	1	no	—
284	#427	C₂²×C₅	1	1	yes	characteristic
285	C₂×#107	—	2	1	no	—
286	C₂×#107	C₂²×C₅	1	1	yes	characteristic
287	C₂×#107	C₂²×C₅	1	1	yes	characteristic
288	#424	C₂²×C₅	1	1	yes	characteristic
289	C₂×#104	C₂²×C₅	1	1	yes	characteristic
290	C₂×#104	C₂²×C₅	1	1	yes	characteristic
291	#418	C₂²×C₅	1	1	yes	characteristic
292	#417	C₂²×C₅	1	1	yes	characteristic
293	#417	C₂²×C₅	1	1	yes	characteristic
294	#417	C₂²×C₅	1	1	yes	characteristic
295	C₂×#102	C₂²×C₅	1	1	yes	characteristic
296	C₂×#102	C₂²×C₅	1	1	yes	characteristic
297	C₂×#100	C₂²×C₅	1	1	yes	characteristic
298	#408	C₂²×C₅	1	1	yes	characteristic
299	#405	C₂²×C₅	1	1	yes	characteristic
300	#390	C₂²×C₅	1	1	yes	characteristic
301	#384	C₂²×C₅	1	1	yes	characteristic
302	#343	C₂²×C₅	1	1	yes	characteristic
303	#343	C₂²×C₅	1	1	yes	characteristic
304	#343	C₂²×C₅	1	1	yes	characteristic
305	#341	C₂²×C₅	1	1	yes	characteristic
306	#341	C₂²×C₅	1	1	yes	characteristic
307	#339	C₂²×C₅	1	1	yes	characteristic
308	#339	C₂²×C₅	1	1	yes	characteristic
309	#336	C₂²×C₅	1	1	yes	characteristic
310	C₂³×C₅	—	2	1	no	—
311	C₂³×C₅	—	2	1	no	—
312	C₂³×C₅	—	4	1	no	—
313	C₂³×C₅	—	4	1	no	—
314	C₂³×C₅	—	4	1	no	—
315	C₂³×C₅	—	8	1	no	—
316	C₂³×C₅	—	8	1	no	—
317	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
318	C₅×Q₈	—	2	1	no	—
319	C₅×Q₈	—	2	1	no	—
320	C₅×Q₈	—	2	1	no	—
321	C₅×Q₈	#123	1	2	yes	—
322	C₅×D₄	—	2	2	no	—
323	C₅×D₄	—	2	2	no	—
324	C₅×D₄	—	2	1	no	—
325	C₅×D₄	—	2	1	no	—
326	C₅×D₄	—	2	1	no	—
327	C₅×D₄	—	4	1	no	—
328	C₅×D₄	—	4	1	no	—
329	C₅×D₄	—	4	1	no	—
330	C₅×D₄	—	4	1	no	—
331	C₅×D₄	—	4	1	no	—
332	C₅×D₄	—	4	1	no	—
333	C₅×D₄	—	16	1	no	—
334	C₅×D₄	—	16	1	no	—
335	C₅×D₄	#123	1	2	yes	—
336	C₂×C₄×C₅	—	2	1	no	—
337	C₂×C₄×C₅	—	2	2	no	—
338	C₂×C₄×C₅	—	2	2	no	—
339	C₂×C₄×C₅	—	2	2	no	—
340	C₂×C₄×C₅	—	2	1	no	—
341	C₂×C₄×C₅	—	2	1	no	—
342	C₂×C₄×C₅	—	2	1	no	—
343	C₂×C₄×C₅	—	2	1	no	—
344	C₂×C₄×C₅	—	2	1	no	—
345	C₂×C₄×C₅	—	4	1	no	—
346	C₂×C₄×C₅	—	4	1	no	—
347	C₂×C₄×C₅	—	4	1	no	—
348	C₂×C₄×C₅	—	4	1	no	—
349	C₂×C₄×C₅	—	4	1	no	—
350	C₂×C₄×C₅	—	4	1	no	—
351	C₂×C₄×C₅	—	4	1	no	—
352	C₂×C₄×C₅	—	8	1	no	—
353	C₂×C₄×C₅	—	8	1	no	—
354	C₂×C₄×C₅	—	8	1	no	—
355	C₂×C₄×C₅	—	16	1	no	—
356	C₂×C₄×C₅	—	16	1	no	—
357	C₂×C₄×C₅	—	16	1	no	—
358	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
359	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
360	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
361	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
362	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
363	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
364	C₅×C₈	—	2	1	no	—
365	C₅×C₈	—	2	1	no	—
366	C₅×C₈	—	2	1	no	—
367	C₅×C₈	—	2	2	no	—
368	C₅×C₈	—	2	1	no	—
369	C₅×C₈	—	4	1	no	—
370	C₅×C₈	—	4	1	no	—
371	C₅×C₈	—	4	1	no	—
372	C₂×#41	C₂³×C₅	1	1	yes	characteristic
373	C₂×#41	C₂³×C₅	1	1	yes	characteristic
374	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
375	AGL(1,ℤ/8)	—	2	2	no	—
376	AGL(1,ℤ/8)	—	2	2	no	—
377	#135	—	2	2	no	—
378	#135	—	2	2	no	—
379	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
380	C₂×SD₁₆	—	2	1	no	—
381	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
382	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
383	C₂×D₈	—	2	1	no	—
384	C₂×D₈	C₅×D₄	1	1	yes	characteristic
385	#131	—	2	2	no	—
386	#131	—	2	2	no	—
387	C₂×M₁₆	—	2	1	no	—
388	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
389	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
390	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
391	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
392	C₂²×C₈	—	2	1	no	—
393	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
394	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
395	#127	—	2	1	no	—
396	#124	—	2	1	no	—
397	#121	—	2	1	no	—
398	#121	—	2	1	no	—
399	#120	—	4	1	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	4	1	no	—
407	C₄×D₄	—	4	1	no	—
408	C₄×D₄	—	4	1	no	—
409	#117	—	2	1	no	—
410	#117	—	2	1	no	—
411	#117	—	2	1	no	—
412	#117	C₂³×C₅	1	1	yes	characteristic
413	#117	C₂³×C₅	1	1	yes	characteristic
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	4	1	no	—
418	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
421	C₂×C₂²⋊C₄	—	4	1	no	—
422	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
423	#107	—	2	1	no	—
424	#107	—	2	1	no	—
425	#107	—	2	1	no	—
426	#107	—	2	1	no	—
427	#107	—	2	1	no	—
428	#107	—	2	1	no	—
429	#107	—	2	1	no	—
430	#107	C₅×D₄	1	1	yes	characteristic
431	#107	C₅×D₄	1	1	yes	characteristic
432	#106	—	2	1	no	—
433	#106	—	2	1	no	—
434	#106	—	2	1	no	—
435	#106	C₅×D₄	1	1	yes	characteristic
436	#106	C₅×D₄	1	1	yes	characteristic
437	C₄⋊C₈	—	2	1	no	—
438	C₄⋊C₈	—	2	1	no	—
439	#104	—	2	2	no	—
440	#104	—	2	2	no	—
441	#103	—	2	1	no	—
442	#103	—	2	1	no	—
443	#103	—	2	1	no	—
444	#102	—	2	1	no	—
445	#102	—	2	1	no	—
446	#102	—	2	1	no	—
447	#102	—	2	1	no	—
448	#102	—	2	1	no	—
449	#102	—	2	1	no	—
450	#102	—	2	1	no	—
451	#102	—	2	1	no	—
452	#102	—	2	1	no	—
453	#100	—	2	2	no	—
454	C₂²⋊C₈	—	2	1	no	—
455	C₂²⋊C₈	—	2	1	no	—
456	#97	—	2	1	no	—
457	C₄×C₈	—	2	1	no	—
458	C₂²×C₅	—	2	1	no	—
459	C₂²×C₅	—	2	1	no	—
460	C₂²×C₅	—	2	2	no	—
461	C₂²×C₅	—	4	1	no	—
462	C₂²×C₅	—	4	1	no	—
463	C₂²×C₅	—	4	1	no	—
464	C₂²×C₅	—	4	1	no	—
465	C₂²×C₅	—	4	1	no	—
466	C₂²×C₅	—	8	2	no	—
467	C₂²×C₅	—	8	1	no	—
468	C₂²×C₅	—	8	1	no	—
469	C₂²×C₅	—	8	1	no	—
470	C₂²×C₅	—	16	1	no	—
471	C₂²×C₅	#539	1	1	yes	characteristic
472	C₂²×C₅	D₄²	1	1	yes	characteristic
473	C₂²×C₅	#553	1	1	yes	characteristic
474	C₄×C₅	—	2	2	no	—
475	C₄×C₅	—	4	2	no	—
476	C₄×C₅	—	4	1	no	—
477	C₄×C₅	—	4	1	no	—
478	C₄×C₅	—	4	1	no	—
479	C₄×C₅	—	4	1	no	—
480	C₄×C₅	—	4	1	no	—
481	C₄×C₅	—	8	1	no	—
482	C₄×C₅	—	8	1	no	—
483	C₄×C₅	—	8	1	no	—
484	C₄×C₅	C₂×#123	1	1	yes	characteristic
485	C₄×C₅	#525	1	1	yes	characteristic
486	C₄×C₅	#547	1	1	yes	characteristic
487	C₄×C₅	#548	1	1	yes	characteristic
488	C₂⁴	—	2	1	no	—
489	#41	—	2	1	no	—
490	#41	—	2	1	no	—
491	#41	—	2	1	no	—
492	#41	—	2	1	no	—
493	#41	—	2	1	no	—
494	#41	—	2	1	no	—
495	#41	C₅×#41	1	2	yes	—
496	#41	C₅×#41	1	2	yes	—
497	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
498	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	4	1	no	—
508	C₂×D₄	—	4	1	no	—
509	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
510	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	4	1	no	—
518	C₂²×C₄	—	4	1	no	—
519	C₂²×C₄	—	4	1	no	—
520	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
521	Q₁₆	—	2	2	no	—
522	SD₁₆	—	2	2	no	—
523	SD₁₆	—	2	2	no	—
524	SD₁₆	—	4	2	no	—
525	D₈	—	2	2	no	—
526	D₈	—	4	2	no	—
527	M₁₆	—	2	4	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	1	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	C₅×#41	1	2	yes	—
536	C₂×C₈	—	2	4	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
546	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
547	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
548	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	4	1	no	—
556	C₄⋊C₄	—	4	1	no	—
557	C₄⋊C₄	—	4	1	no	—
558	C₄⋊C₄	—	8	2	no	—
559	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
560	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
561	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
562	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	4	1	no	—
566	C₂²⋊C₄	—	4	1	no	—
567	C₂²⋊C₄	—	4	1	no	—
568	C₂²⋊C₄	—	4	1	no	—
569	C₂²⋊C₄	—	4	1	no	—
570	C₂²⋊C₄	—	8	2	no	—
571	C₄²	—	2	1	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	2	1	no	—
574	C₄²	—	2	1	no	—
575	C₄²	—	4	1	no	—
576	C₄²	—	4	1	no	—
577	C₄²	—	4	1	no	—
578	C₂×C₅	—	2	1	no	—
579	C₂×C₅	—	2	1	no	—
580	C₂×C₅	—	4	2	no	—
581	C₂×C₅	—	8	1	no	—
582	C₂×C₅	—	16	1	no	—
583	C₂×C₅	#2443	1	1	yes	characteristic
584	C₂×C₅	#3289	1	1	yes	characteristic
585	C₂×C₅	#3292	1	1	yes	characteristic
586	C₂³	—	2	1	no	—
587	C₂³	—	2	1	no	—
588	C₂³	—	4	1	no	—
589	C₂³	—	4	1	no	—
590	C₂³	—	4	1	no	—
591	C₂³	—	8	1	no	—
592	C₂³	—	8	1	no	—
593	C₂³	C₅×H(5,𝔽₂)	1	1	yes	characteristic
594	Q₈	—	2	1	no	—
595	Q₈	—	2	1	no	—
596	Q₈	—	2	1	no	—
597	Q₈	C₅×#123	1	2	yes	—
598	D₄	—	2	2	no	—
599	D₄	—	2	2	no	—
600	D₄	—	2	1	no	—
601	D₄	—	2	1	no	—
602	D₄	—	2	1	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	1	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	1	no	—
607	D₄	—	4	1	no	—
608	D₄	—	4	1	no	—
609	D₄	—	16	1	no	—
610	D₄	—	16	1	no	—
611	D₄	C₅×#123	1	2	yes	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	8	1	no	—
629	C₂×C₄	—	8	1	no	—
630	C₂×C₄	—	8	1	no	—
631	C₂×C₄	—	16	1	no	—
632	C₂×C₄	—	16	1	no	—
633	C₂×C₄	—	16	1	no	—
634	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
635	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
636	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
637	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
638	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
639	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
640	C₈	—	2	1	no	—
641	C₈	—	2	1	no	—
642	C₈	—	2	1	no	—
643	C₈	—	2	2	no	—
644	C₈	—	2	1	no	—
645	C₈	—	4	1	no	—
646	C₈	—	4	1	no	—
647	C₈	—	4	1	no	—
648	C₅	#26190	1	1	yes	characteristic, direct product factor, Sylow
649	C₂²	—	2	1	no	—
650	C₂²	—	2	1	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	1	no	—
655	C₂²	—	4	1	no	—
656	C₂²	—	4	1	no	—
657	C₂²	—	8	2	no	—
658	C₂²	—	8	1	no	—
659	C₂²	—	8	1	no	—
660	C₂²	—	8	1	no	—
661	C₂²	—	16	1	no	—
662	C₂²	C₅×#539	1	1	yes	characteristic
663	C₂²	C₅×D₄²	1	1	yes	characteristic
664	C₂²	C₅×#553	1	1	yes	characteristic
665	C₄	—	2	2	no	—
666	C₄	—	4	2	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	1	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	8	1	no	—
673	C₄	—	8	1	no	—
674	C₄	—	8	1	no	—
675	C₄	C₂×C₅×#123	1	1	yes	characteristic
676	C₄	C₅×#525	1	1	yes	characteristic
677	C₄	C₅×#547	1	1	yes	characteristic
678	C₄	C₅×#548	1	1	yes	characteristic
679	C₂	—	2	1	no	—
680	C₂	—	2	1	no	—
681	C₂	—	4	2	no	—
682	C₂	—	8	1	no	—
683	C₂	—	16	1	no	—
684	C₂	C₅×#2443	1	1	yes	characteristic
685	C₂	C₅×#3289	1	1	yes	characteristic
686	C₂	C₅×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₅×#3045	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅×#3096	C₂	semidirect
3. C₅×#2951	C₂	semidirect
4. C₅×#2913	C₂	semidirect
5. C₅×#2093	C₂	semidirect
6. C₅×#2089	C₂	semidirect
7. C₅×#2078	C₂	semidirect
8. C₅×#2037	C₂	semidirect
9. C₅×#1931	C₂	semidirect
10. C₅×#1922	C₂	semidirect
11. C₅×#1907	C₂	semidirect
12. C₅×#1883	C₂	semidirect
13. C₅×#1876	C₂	semidirect
14. C₅×#1859	C₂	semidirect
15. C₅×#1859	C₂	semidirect
67. #26190	C₅	direct (N × H)
154. #3096	C₂×C₅	semidirect
156. #2951	C₂×C₅	semidirect
157. #2913	C₂×C₅	semidirect
158. #2093	C₂×C₅	semidirect
159. #2089	C₂×C₅	semidirect
160. #2078	C₂×C₅	semidirect
161. #2037	C₂×C₅	semidirect
162. #1931	C₂×C₅	semidirect
163. #1922	C₂×C₅	semidirect
164. #1907	C₂×C₅	semidirect
165. #1883	C₂×C₅	semidirect
166. #1876	C₂×C₅	semidirect
167. #1859	C₂×C₅	semidirect
168. #1859	C₂×C₅	semidirect
648. C₅	#26190	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#3096	2
C₅×#3045	2
C₅×#2951	2
C₅×#2913	2
C₅×#2093	2
C₅×#2089	2
C₅×#2078	2
C₅×#2037	2
C₅×#1931	2
C₅×#1922	2
C₅×#1907	2
C₅×#1883	2
C₅×#1876	2
C₅×#1859	2
#26190	5

Subgroups — C5×#26190 #11948910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#26190 #11948910